Integrand size = 43, antiderivative size = 305 \[ \int \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {2 \left (18 a A b+9 a^2 B+7 b^2 B+14 a b C\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 \left (110 a b B+11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{231 d}+\frac {2 \left (110 a b B+11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{231 d}+\frac {2 \left (18 a A b+9 a^2 B+7 b^2 B+14 a b C\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac {2 \left (11 A b^2+22 a b B+4 a^2 C+9 b^2 C\right ) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{77 d}+\frac {2 b (11 b B+4 a C) \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac {2 C \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{11 d} \]
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Time = 0.68 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.163, Rules used = {3128, 3112, 3102, 2827, 2715, 2720, 2719} \[ \int \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {2 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (11 a^2 (7 A+5 C)+110 a b B+5 b^2 (11 A+9 C)\right )}{231 d}+\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (9 a^2 B+18 a A b+14 a b C+7 b^2 B\right )}{15 d}+\frac {2 \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (4 a^2 C+22 a b B+11 A b^2+9 b^2 C\right )}{77 d}+\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (9 a^2 B+18 a A b+14 a b C+7 b^2 B\right )}{45 d}+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)} \left (11 a^2 (7 A+5 C)+110 a b B+5 b^2 (11 A+9 C)\right )}{231 d}+\frac {2 b (4 a C+11 b B) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{99 d}+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^2}{11 d} \]
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Rule 2715
Rule 2719
Rule 2720
Rule 2827
Rule 3102
Rule 3112
Rule 3128
Rubi steps \begin{align*} \text {integral}& = \frac {2 C \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{11 d}+\frac {2}{11} \int \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x)) \left (\frac {1}{2} a (11 A+5 C)+\frac {1}{2} (11 A b+11 a B+9 b C) \cos (c+d x)+\frac {1}{2} (11 b B+4 a C) \cos ^2(c+d x)\right ) \, dx \\ & = \frac {2 b (11 b B+4 a C) \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac {2 C \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{11 d}+\frac {4}{99} \int \cos ^{\frac {3}{2}}(c+d x) \left (\frac {9}{4} a^2 (11 A+5 C)+\frac {11}{4} \left (18 a A b+9 a^2 B+7 b^2 B+14 a b C\right ) \cos (c+d x)+\frac {9}{4} \left (11 A b^2+22 a b B+4 a^2 C+9 b^2 C\right ) \cos ^2(c+d x)\right ) \, dx \\ & = \frac {2 \left (11 A b^2+22 a b B+4 a^2 C+9 b^2 C\right ) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{77 d}+\frac {2 b (11 b B+4 a C) \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac {2 C \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{11 d}+\frac {8}{693} \int \cos ^{\frac {3}{2}}(c+d x) \left (\frac {9}{8} \left (110 a b B+11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right )+\frac {77}{8} \left (18 a A b+9 a^2 B+7 b^2 B+14 a b C\right ) \cos (c+d x)\right ) \, dx \\ & = \frac {2 \left (11 A b^2+22 a b B+4 a^2 C+9 b^2 C\right ) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{77 d}+\frac {2 b (11 b B+4 a C) \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac {2 C \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{11 d}+\frac {1}{9} \left (18 a A b+9 a^2 B+7 b^2 B+14 a b C\right ) \int \cos ^{\frac {5}{2}}(c+d x) \, dx+\frac {1}{77} \left (110 a b B+11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \int \cos ^{\frac {3}{2}}(c+d x) \, dx \\ & = \frac {2 \left (110 a b B+11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{231 d}+\frac {2 \left (18 a A b+9 a^2 B+7 b^2 B+14 a b C\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac {2 \left (11 A b^2+22 a b B+4 a^2 C+9 b^2 C\right ) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{77 d}+\frac {2 b (11 b B+4 a C) \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac {2 C \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{11 d}+\frac {1}{15} \left (18 a A b+9 a^2 B+7 b^2 B+14 a b C\right ) \int \sqrt {\cos (c+d x)} \, dx+\frac {1}{231} \left (110 a b B+11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx \\ & = \frac {2 \left (18 a A b+9 a^2 B+7 b^2 B+14 a b C\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 \left (110 a b B+11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{231 d}+\frac {2 \left (110 a b B+11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{231 d}+\frac {2 \left (18 a A b+9 a^2 B+7 b^2 B+14 a b C\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac {2 \left (11 A b^2+22 a b B+4 a^2 C+9 b^2 C\right ) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{77 d}+\frac {2 b (11 b B+4 a C) \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac {2 C \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{11 d} \\ \end{align*}
Time = 3.21 (sec) , antiderivative size = 239, normalized size of antiderivative = 0.78 \[ \int \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {154 \left (9 a^2 B+7 b^2 B+2 a b (9 A+7 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+10 \left (110 a b B+11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+\frac {1}{12} \sqrt {\cos (c+d x)} \left (154 \left (72 a A b+36 a^2 B+43 b^2 B+86 a b C\right ) \cos (c+d x)+5 \left (3432 a b B+132 a^2 (14 A+13 C)+3 b^2 (572 A+531 C)+36 \left (11 A b^2+22 a b B+11 a^2 C+16 b^2 C\right ) \cos (2 (c+d x))+154 b (b B+2 a C) \cos (3 (c+d x))+63 b^2 C \cos (4 (c+d x))\right )\right ) \sin (c+d x)}{1155 d} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(862\) vs. \(2(333)=666\).
Time = 15.55 (sec) , antiderivative size = 863, normalized size of antiderivative = 2.83
method | result | size |
default | \(\text {Expression too large to display}\) | \(863\) |
parts | \(\text {Expression too large to display}\) | \(1060\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.15 (sec) , antiderivative size = 358, normalized size of antiderivative = 1.17 \[ \int \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {2 \, {\left (315 \, C b^{2} \cos \left (d x + c\right )^{4} + 385 \, {\left (2 \, C a b + B b^{2}\right )} \cos \left (d x + c\right )^{3} + 165 \, {\left (7 \, A + 5 \, C\right )} a^{2} + 1650 \, B a b + 75 \, {\left (11 \, A + 9 \, C\right )} b^{2} + 45 \, {\left (11 \, C a^{2} + 22 \, B a b + {\left (11 \, A + 9 \, C\right )} b^{2}\right )} \cos \left (d x + c\right )^{2} + 77 \, {\left (9 \, B a^{2} + 2 \, {\left (9 \, A + 7 \, C\right )} a b + 7 \, B b^{2}\right )} \cos \left (d x + c\right )\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - 15 \, \sqrt {2} {\left (11 i \, {\left (7 \, A + 5 \, C\right )} a^{2} + 110 i \, B a b + 5 i \, {\left (11 \, A + 9 \, C\right )} b^{2}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 15 \, \sqrt {2} {\left (-11 i \, {\left (7 \, A + 5 \, C\right )} a^{2} - 110 i \, B a b - 5 i \, {\left (11 \, A + 9 \, C\right )} b^{2}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 231 \, \sqrt {2} {\left (-9 i \, B a^{2} - 2 i \, {\left (9 \, A + 7 \, C\right )} a b - 7 i \, B b^{2}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 231 \, \sqrt {2} {\left (9 i \, B a^{2} + 2 i \, {\left (9 \, A + 7 \, C\right )} a b + 7 i \, B b^{2}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right )}{3465 \, d} \]
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Timed out. \[ \int \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\text {Timed out} \]
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\[ \int \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{2} \cos \left (d x + c\right )^{\frac {3}{2}} \,d x } \]
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\[ \int \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{2} \cos \left (d x + c\right )^{\frac {3}{2}} \,d x } \]
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Time = 3.72 (sec) , antiderivative size = 401, normalized size of antiderivative = 1.31 \[ \int \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {2\,A\,a^2\,\left (\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )+\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )\right )}{3\,d}-\frac {2\,B\,a^2\,{\cos \left (c+d\,x\right )}^{7/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {11}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,A\,b^2\,{\cos \left (c+d\,x\right )}^{9/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {9}{4};\ \frac {13}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{9\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,C\,a^2\,{\cos \left (c+d\,x\right )}^{9/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {9}{4};\ \frac {13}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{9\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,B\,b^2\,{\cos \left (c+d\,x\right )}^{11/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {11}{4};\ \frac {15}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{11\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,C\,b^2\,{\cos \left (c+d\,x\right )}^{13/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {13}{4};\ \frac {17}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{13\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {4\,A\,a\,b\,{\cos \left (c+d\,x\right )}^{7/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {11}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {4\,B\,a\,b\,{\cos \left (c+d\,x\right )}^{9/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {9}{4};\ \frac {13}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{9\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {4\,C\,a\,b\,{\cos \left (c+d\,x\right )}^{11/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {11}{4};\ \frac {15}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{11\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}} \]
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